Modal Analysis
Modal analysis determines the undamped free-vibration characteristics of a structure. It solves the eigenvalue problem ([K] − ωᵢ²[M]){φᵢ} = {0}, where [K] is the stiffness matrix, [M] is the mass matrix, ωᵢ is the i-th angular natural frequency, and {φᵢ} is its mode shape. A nonzero mode shape is possible only when the determinant of ([K] − ωᵢ²[M]) is zero. For a single-degree-of-freedom system, ωₙ = √(k/m), showing that increasing stiffness raises the natural frequency while increasing mass lowers it. Mode shapes describe relative deformation and may be scaled or normalized; they are not unique deformation magnitudes. Boundary conditions strongly affect the result: a free-free three-dimensional body has six rigid-body modes, while added constraints generally raise flexible-mode frequencies. Sufficient modes should be retained to cover the excitation frequency range or provide adequate effective mass participation in the directions of interest.
Knowledge Check
1.For ([K] - ωᵢ²[M]){φᵢ} = {0} to have a nonzero mode shape, what condition must be satisfied?
2.A single-degree-of-freedom model has k = 100 N/m and m = 4 kg. What is its angular natural frequency?
3.What is the natural frequency in hertz for the same model with ωₙ = 5 rad/s?
4.Which operation changes the physical mode shape of an eigenmode?
5.If the stiffness of a single-degree-of-freedom system is quadrupled while its mass remains unchanged, what happens to its natural frequency?
6.If the mass of a system doubles while its stiffness remains unchanged, what happens to its natural frequency?
7.How many rigid-body modes does a fully unconstrained three-dimensional solid body have?
8.What is the usual effect of adding displacement constraints to an FEA modal model?
9.What is an appropriate criterion for deciding how many modes to extract?
10.What does modal analysis indicate about harmonic response near a natural frequency?